Keywords

1 Introduction

Errors always happen in our life, work and study, sometimes with huge destructive effect, so it is necessary to study how to avoid and eliminate errors. And in order to avoid and eliminate errors, we have to study causes and laws of errors. And this paper discusses how to utilize the error matrix equation to eliminate errors.

2 The Concept of Error Matrix

Definition 1

Let

$$ \left( {\begin{array}{*{20}c} {\left( {\left( {\begin{array}{*{20}c} {U_{11} } & {U_{12} } & \cdots & {U_{1k} } \\ \end{array} } \right),\;x_{11} } \right)} & \cdots & {\left( {\left( {\begin{array}{*{20}c} {U_{11} } & {U_{12} } & \cdots & {U_{1k} } \\ \end{array} } \right),\;x_{1n} } \right)} \\ {\left( {\left( {\begin{array}{*{20}c} {U_{21} } & {U_{22} } & \cdots & {U_{2k} } \\ \end{array} } \right),\;x_{21} } \right)} & \cdots & {\left( {\left( {\begin{array}{*{20}c} {U_{21} } & {U_{22} } & \cdots & {U_{2k} } \\ \end{array} } \right),\;x_{2n} } \right)} \\ \cdots & \cdots & \cdots \\ {\left( {\left( {\begin{array}{*{20}c} {U_{m1} } & {U_{m2} } & \cdots & {U_{mk} } \\ \end{array} } \right),\;x_{m1} } \right)} & \cdots & {\left( {\left( {\begin{array}{*{20}c} {U_{m1} } & {U_{m2} } & \cdots & {U_{mk} } \\ \end{array} } \right),\;x_{mn} } \right)} \\ \end{array} } \right) $$

be an m × n error matrix of K elements.

Definition 2

Let

$$ \left( {\begin{array}{*{20}c} {U_{20} } & {S_{20} } & {\vec{P}_{20} } & {T_{20} } & {L_{20} } & {y_{20} } & {G_{u20} } \\ {U_{21} } & {S_{21} } & {\vec{P}_{21} } & {T_{21} } & {L_{21} } & {y_{21} } & {G_{u21} } \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ {U_{2t} } & {S_{2t} } & {\vec{P}_{2t} } & {T_{2t} } & {L_{2t} } & {y_{2t} } & {G_{u2t} } \\ \end{array} } \right) $$

be a (t + 1) × 7 or m × 7 error matrix. Each element of this kind of error matrix is called a set.

Definition 3

The set relationship containing unknown sets is called set equations.

Definition 4

Let X, A′ and B be m × 7 error matrixes, so XA′ ⊇ (or other relational operators) B is named as set (matrix) equations.

3 Error Matrix Equation

Error matrix equation:

$$ \begin{aligned} & {\text{Type}}\;{\text{I}}\;{\text{AX}}\; \supseteq \;{\text{B}} \,{\text{A}} \cdot {\text{X}}\; \supseteq \;{\text{B}} \,{\text{A}}{\blacktriangle }{\text{X}}\; \supseteq \;{\text{B}} \,{\text{A}} \vee {\text{X}}\; \supseteq \;{\text{B}} {\text{A}} \wedge {\text{X }}\; \supseteq \;{\text{B}} \\ & {\text{Type}}\;{\text{II}}\;{\text{XA}}\; \supseteq \;{\text{B}}\, {\text{X}} \cdot {\text{A}}\; \supseteq \;{\text{B}}\, {\text{X}}{\blacktriangle }{\text{A}}\; \supseteq \;{\text{B}}\,{\text{X}} \vee {\text{A}}\; \supseteq \;{\text{B}} {\text{X}} \wedge {\text{A}}\; \supseteq \;{\text{B}} \\ \end{aligned} $$

4 The Solution of Error Matrix Equation

The solution of Type II 1 XA = B

$$ \begin{aligned} X & = (u,v) = (U_{1} ,S_{1} ,\vec{P}_{1} ,T_{1} ,L_{1} ,x) = \left( {\begin{array}{*{20}c} {U_{10x} } & {S_{10x} } & {\vec{P}_{10x} } & {T_{10x} } & {L_{10x} } & {x_{10x} } & {G_{10x} } \\ {U_{11x} } & {S_{11x} } & {\vec{P}_{11x} } & {T_{11x} } & {L_{11x} } & {x_{11x} } & {G_{11x} } \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ {U_{1tx} } & {S_{1tx} } & {\overrightarrow {{P_{1tx} }} } & {T_{1tx} } & {L_{1tx} } & {x_{1tx} } & {G_{1tx} } \\ \end{array} } \right) \\ A & = (U_{2} ,S_{2} ,\vec{P}_{2} ,T_{2} ,L_{2} ,x_{2} ) = \left( {\begin{array}{*{20}c} {U_{20} } & {S_{20} } & {\vec{P}_{20} } & {T_{20} } & {L_{20} } & {y_{20} } & {G_{u20} } \\ {U_{21} } & {S_{21} } & {\vec{P}_{21} } & {T_{21} } & {L_{21} } & {y_{21} } & {G_{u21} } \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ {U_{2t} } & {S_{2t} } & {\vec{P}_{2t} } & {T_{2t} } & {L_{2t} } & {y_{2t} } & {G_{u2t} } \\ \end{array} } \right) \\ & \left( {\begin{array}{*{20}c} {V_{201} } & {S_{v201} } & {\vec{P}_{v201} } & {T_{v201} } & {L_{v201} } & {y_{v201} } & {G_{v201} } \\ {U_{21j} } & {S_{v21j} } & {\vec{P}_{v21j} } & {T_{v21j} } & {L_{v21j} } & {y_{v21j} } & {G_{v21j} } \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ {U_{2m2m1} } & {S_{v2m2m1} } & {\vec{P}_{v2m2m1} } & {T_{v2m2m1} } & {L_{v2m2m1} } & {y_{v2m2m1} } & {G_{v2m2m1} } \\ \end{array} } \right) \\ B & = \left( {\begin{array}{*{20}c} {(b_{11} ,y_{11} )} & {(b_{12} ,y_{12} )} & \cdots & {(b_{1m} ,y_{1m} )} \\ {(b_{21} ,y_{21} )} & {(b_{21} ,y_{21} )} & \cdots & {(b_{2m} ,y_{2m} )} \\ \cdots & \cdots & \cdots & \cdots \\ {(b_{m1} ,y_{m1} )} & {(b_{m1} ,y_{m1} )} & \cdots & {(b_{mm} ,y_{mm} )} \\ \end{array} } \right) \\ \end{aligned} $$

Definition 5

Let

$$ \begin{aligned} XA^{{\prime }} \supseteq & = \left( {\begin{array}{*{20}c} {(w_{11} ,z_{11} )} & {(w_{12} ,z_{12} )} & \cdots & {(w_{1m} ,z_{1m} )} \\ {(w_{21} ,z_{21} )} & {(w_{22} ,z_{22} )} & \cdots & {(w_{2m} ,z_{2m} )} \\ \cdots & \cdots & \cdots & \cdots \\ {(w_{m1} ,z_{m1} )} & {(w_{m2} ,z_{m2} )} & \cdots & {(w_{mm} ,z_{mm} )} \\ \end{array} } \right) \\ & = \left( {\begin{array}{*{20}c} {V_{201} } & {S_{v201} } & {\overrightarrow {P}_{v201} } & {T_{v201} } & {L_{v201} } & {y_{v201} } & {G_{v201} } \\ {V_{21j} } & {S_{v21j} } & {\overrightarrow {P}_{v21j} } & {T_{v21j} } & {L_{v21j} } & {y_{v21j} } & {G_{v21j} } \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ {V_{2m2m1} } & {S_{v2m2m1} } & {\overrightarrow {P}_{v2m2m1} } & {T_{v2m2m1} } & {L_{v2m2m1} } & {y_{v2m2m1} } & {G_{v2m2m1} } \\ \end{array} } \right) \\ \end{aligned} $$

and,

$$ \left( {w_{ij} ,z_{ij} } \right) = \left( {\begin{array}{*{20}c} {U_{1ix} \wedge U_{2j} } & {S_{1ix} \wedge S_{2j} } & {\overrightarrow {P}_{1ix} \wedge \overrightarrow {P}_{2j} } & {T_{1ix} \wedge T_{2j} } & {L_{1ix} \wedge L_{2j} } & {x_{1ix} \wedge y_{2j} } & {G_{U1ix} \wedge G_{U2j} } \\ \end{array} } \right) $$

So the following equation is hold.

$$ \begin{aligned} & \left( {\begin{array}{*{20}c} {U_{10x} \wedge U_{20} } & {S_{10x} \wedge S_{20} } & {\overrightarrow {P}_{10x} \wedge \overrightarrow {P}_{20} } & {T_{10x} \wedge T_{20} } & {L_{10x} \wedge L_{20} } & {x_{10x} \wedge y_{20} } & {G_{U10x} \wedge G_{U20} } \\ {U_{11x} \wedge U_{21} } & {S_{11x} \wedge S_{21} } & {\overrightarrow {P}_{11x} \wedge \overrightarrow {P}_{21} } & {T_{11x} \wedge T_{21} } & {L_{11x} \wedge L_{21} } & {x_{11x} \wedge y_{21} } & {G_{U11x} \wedge G_{U21} } \\ \cdots & \cdots & \cdots & {} & \cdots & \cdots & \cdots \\ {U_{1tx} \wedge U_{2t} } & {S_{1tx} \wedge S_{2t} } & {\overrightarrow {P}_{1tx} \wedge \overrightarrow {P}_{2t} } & {T_{1tx} \wedge T_{2t} } & {L_{1tx} \wedge L_{2t} } & {x_{1tx} \wedge y_{2t} } & {G_{U1tx} \wedge G_{U2t} } \\ \end{array} } \right) \\ & \quad = \left( {\begin{array}{*{20}c} {V_{201} } & {S_{v201} } & {\overrightarrow {P}_{v201} } & {T_{v201} } & {L_{v201} } & {y_{v201} } & {G_{v101} } \\ {V_{21j} } & {S_{v21j} } & {\overrightarrow {P}_{v21j} } & {T_{v21j} } & {L_{v21j} } & {y_{v21j} } & {G_{v11j} } \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ {V_{2m2m1} } & {S_{v2m2m1} } & {\overrightarrow {P}_{v2m2m1} } & {T_{v2m2m1} } & {L_{v2m2m1} } & {y_{v2m2m1} } & {G_{v1m2m1} } \\ \end{array} } \right) \\ \end{aligned} $$

By the definition of equal matrices:if two matrices contain each other,so corresponding elements in both matrices contain each other. So \( \left( {w_{ij} ,z_{ij} } \right) \supseteq \left( {b_{ij} ,y_{ij} } \right) \),

$$ \begin{aligned} & \left( {\begin{array}{*{20}c} {U_{1ix} \wedge U_{2j} } & {S_{1ix} \wedge S_{2j} } & {\overrightarrow {P}_{1ix} \wedge \overrightarrow {P}_{2j} } & {T_{1ix} \wedge T_{2j} } & {L_{1ix} \wedge L_{2j} } & {x_{1ix} \wedge y_{2j} } & {G_{U1ix} \wedge G_{U2j} } \\ \end{array} } \right) \supseteq \left( {b_{ij} ,y_{ij} } \right) \\ & \quad = \left( {\begin{array}{*{20}c} {V_{2ij} } & {S_{v2ij} } & {\overrightarrow {P}_{v2ij} } & {T_{v2ij} } & {L_{v2ij} } & {y_{v2ij} } & {G_{v2ij} } \\ \end{array} } \right) \\ \end{aligned} $$

So the following set equations are obtained:

$$ \begin{array}{*{20}c} {U_{10x} \wedge U_{20} \supseteq V_{v20} } \\ {S_{10x} \wedge S_{20} \supseteq S_{v20} } \\ {\overrightarrow {P}_{10x} \wedge \overrightarrow {P}_{20} \supseteq \overrightarrow {P}_{v20} } \\ {T_{10x} \wedge T_{20} \supseteq T_{v20} } \\ {L_{10x} \wedge L_{20} \supseteq L_{v20} } \\ {x_{10x} \wedge y_{20} \supseteq y_{v20} } \\ {G_{U10x} \wedge G_{U20} \supseteq G_{v20} } \\ { \ldots \ldots } \\ {U_{1ix} \wedge U_{2j} \supseteq V_{v2j} } \\ {S_{1ix} \wedge S_{2j} \supseteq S_{v2j} } \\ {\overrightarrow {P}_{1ix} \wedge \overrightarrow {P}_{2j} \supseteq \overrightarrow {P}_{v2j} } \\ {T_{1ix} \wedge T_{2j} \supseteq T_{v2j} } \\ {L_{1ix} \wedge L_{2j} \supseteq L_{v2j} } \\ {x_{1ix} \wedge y_{2j} \supseteq y_{v2j} } \\ {G_{U1ix} \wedge G_{U2j} \supseteq G_{v2j} } \\ { \ldots \ldots } \\ {U_{ttx} \wedge U_{2t} \supseteq V_{v2t} } \\ {S_{ttx} \wedge S_{2t} \supseteq S_{v2t} } \\ {\overrightarrow {P}_{ttx} \wedge \overrightarrow {P}_{2t} \supseteq \overrightarrow {P}_{v2t} } \\ {T_{ttx} \wedge T_{2t} \supseteq T_{v2t} } \\ {L_{ttx} \wedge L_{2t} \supseteq L_{v2t} } \\ {x_{ttx} \wedge y_{2t} \supseteq y_{v2t} } \\ {G_{Uttx} \wedge G_{U2t} \supseteq G_{v2t} } \\ \end{array} $$

About the operation symbol “∧”, if both sides of the equation are sets, then it means “intersection”, if both sides of the equation are numbers, then it means “minimum”.

As for (U 1ix ∧ U 2j) h 1 (S 1ix ∧ S 2j) h 2 (\( \overrightarrow {P}_{1ix} \wedge \overrightarrow {P}_{2j} \)) ∨ h 3 (T 1ix ∧ T 2j) h 4 (L 1ix ∧ L 2j) h 5 (x 1ix ∧ y 2j) h 6 (G U1ix ∧ G U2j), with “hi, i = 1, 2, …6”, means that elements have been computed could “compose” a complete matrix element (proposition). The mode of combination depends on different situations. One way is to constitute a new set of error elements or error logic proposition by parameter. And this way is called the multiplication of m × 7 error matrix.

Since what we required in solving practical problems is not XiA′ = B, but XiA′ ⊇ B, So we find a more general model of error matrix equation, that is to find out the solution of type II 1 equation XA′ ⊇ B.

Theorem 1

The sufficient and necessary condition for the solvability of error matrix equation XA′ ⊇ B is the solvability of X i A′ ⊇ B i , i = (1, 2, ……, m2).

Proof Suppose XA′ ⊇ B has solvability, it is can be known by the definitions of XA′ ⊇ B and XiA′ ⊇ Bi, i = (1, 2, …, m2) that they are the equivalent equations, so it is necessary for XiA′ ⊇ Bi, i = (1, 2, …, m2) has solvability; Otherwise, if the solvability of XiA′ ⊇ Bi, i = (1, 2, …, m2) exists, similarly it does for XA′ ⊇ B.

Thereout, we use the method of discussing the solvability of XiA′ = Bi, i = (0, 1, 2, …, m2) to discuss the solution of XA′ = B.

Then in XiA′ ⊇ Bi, we can get

$$ \begin{aligned} & \left( {\begin{array}{*{20}c} {U_{1ix} } & {S_{1ix} } & {\overrightarrow {P}_{1ix} } & {T_{1ix} } & {L_{1ix} } & {v_{1ix} } & {G_{u1ix} } \\ \end{array} } \right)A^{\prime} \supseteq \\ & \left( {U_{1ix} \wedge U_{20} } \right) \vee \left( {S_{1ix} \wedge S_{20} } \right) \vee \left( {\overrightarrow {P}_{1ix} \wedge \overrightarrow {P}_{20} } \right) \vee \left( {T_{1ix} \wedge T_{20} } \right) \vee \left( {L_{1ix} \wedge L_{20} } \right) \\ &\vee \left( {x_{1ix} \wedge x_{20} } \right) \vee \left( {G_{U1ix} \wedge G_{U20} } \right) \\ & \ldots \ldots \\ & \left( {U_{1ix} \wedge U_{2j} } \right) \vee \left( {S_{1ix} \wedge S_{2j} } \right) \vee \left( {\overrightarrow {P}_{1ix} \wedge \overrightarrow {P}_{2j} } \right) \vee \left( {T_{1ix} \wedge T_{2j} } \right) \vee \left( {L_{1ix} \wedge L_{2j} } \right) \\ &\vee \left( {x_{1ix} \wedge x_{2j} } \right) \vee \left( {G_{U1ix} \wedge G_{U2j} } \right) \\ & \ldots \ldots \\ & \left( {U_{1ix} \wedge U_{2m1} } \right) \vee \left( {S_{1ix} \wedge S_{2m1} } \right) \vee \left( {\overrightarrow {P}_{1ix} \wedge \overrightarrow {P}_{2m1} } \right) \vee \left( {T_{1ix} \wedge T_{2m1} } \right) \\ &\vee \left( {L_{1ix} \wedge L_{2m1} } \right) \vee \left( {x_{1ix} \wedge x_{2m1} } \right) \vee \left( {G_{U1ix} \wedge G_{U2m1} } \right) \\ & \quad \supseteq \left( {\begin{array}{*{20}c} {\left( {b_{i1} ,y_{i1} } \right)} & {\left( {b_{i2} ,y_{i2} } \right)} & \cdots & {\left( {b_{im1} ,y_{im1} } \right)} \\ \end{array} } \right) \\ \end{aligned} $$

Namely,

$$ \begin{aligned} & \left({U_{1ix} \; \wedge \;U_{20} } \right)\; \vee \;\left( {S_{1ix} \; \wedge \;S_{20} } \right)\; \vee \;\left( {\overrightarrow {P}_{1ix} \; \wedge \;\overrightarrow {P}_{20} } \right)\; \vee \;\left( {T_{1ix} \; \wedge \;T_{20} } \right)\\ &\vee \;\left({L_{1ix} \; \wedge \;L_{20} } \right)\; \vee \;\left( {x_{1ix} \; \wedge \;x_{20} } \right)\; \vee \;\left( {G_{U1ix} \; \wedge \;G_{U20} } \right)\; \supseteq \\ & \left( {\begin{array}{*{20}c} {V_{20} } & {S_{v20} } & {\overrightarrow {P}_{v20} } & {T_{v20} } & {L_{v20} } & {y_{v20} } & {G_{v20} } \\ \end{array} } \right) \\ & \ldots \ldots \\ & \left( {U_{1ix} \; \wedge \;U_{2j} } \right)\; \vee \;\left( {S_{1ix} \; \wedge \;S_{2j} } \right)\; \vee \;\left( {\overrightarrow {P}_{1ix} \; \wedge \;\overrightarrow {P}_{2j} } \right)\; \vee \;\left( {T_{1ix} \; \wedge \;T_{2j} } \right)\; \vee \,\left( {L_{1ix} \; \wedge \;L_{2j} } \right)\\ & \vee \;\left({x_{1ix} \; \wedge \;x_{2j} } \right)\; \vee \;\left( {G_{U1ix} \; \wedge \;G_{U2j} } \right)\; \supseteq \\ & \left( {\begin{array}{*{20}c} {V_{2j} } & {S_{v2j} } & {\overrightarrow {P}_{v2j} } & {T_{v2j} } & {L_{v2j} } & {y_{v2j} } & {G_{v2j} } \\ \end{array} } \right) \\ & \ldots \ldots \\ & \left( {U_{1tx} \; \wedge \;U_{2m1} } \right)\, \vee \;\left( {S_{1tx} \; \wedge \,S_{2m1} } \right)\; \vee \;\left( {\overrightarrow {P}_{1tx} \; \wedge \,\overrightarrow {P}_{2m1} } \right)\; \vee \;\left( {T_{1tx} \, \wedge \;T_{2m1} } \right)\; \vee \,\left( {L_{1tx} \, \wedge \;L_{2m1} } \right)\\ & \vee \;\left({x_{1tx} \; \wedge \,x_{2m1} } \right)\; \vee \;\left( {G_{U1tx} \; \wedge \;G_{U2m1} } \right) \supseteq \\ & \left( {\begin{array}{*{20}c} {V_{2t} } & {S_{v2t} } & {\overrightarrow {P}_{v2t} } & {T_{v2t} } & {L_{v2t} } & {y_{v2t} } & {G_{v2t} } \\ \end{array} } \right) \\ \end{aligned} $$

A series of set equations be obtained:

$$ \begin{array}{*{20}c} {\left( {U_{1ix} \; \wedge \;U_{20} } \right)\; \supseteq \;V_{20} } \\ {\left( {S_{1ix} \; \wedge \;S_{20} } \right)\; \supseteq \;S_{v20} } \\ {\left( {\overrightarrow {P}_{1ix} \; \wedge \,\overrightarrow {P}_{20} } \right)\; \supseteq \;\overrightarrow {P}_{v20} } \\ {\left( {T_{1ix} \; \wedge \;T_{20} } \right)\, \supseteq \,T_{v20} } \\ {\left( {L_{1ix} \; \wedge \;L_{20} } \right)\; \supseteq \;L_{v20} } \\ {\left( {x_{1ix} \; \wedge \;x_{20} } \right)\; \supseteq \;y_{v20} } \\ {\left( {G_{U1ix} \, \wedge \;G_{U20} } \right)\; \supseteq \;G_{v20} } \\ { \ldots \ldots } \\ {\left( {U_{1ix} \; \wedge \,U_{2j} } \right)\; \supseteq \;V_{2j} } \\ {\left( {S_{1ix} \; \wedge \;S_{2j} } \right)\, \supseteq \;S_{v2j} } \\ {\left( {\overrightarrow {P}_{1ix} \; \wedge \;\overrightarrow {P}_{2j} } \right)\; \supseteq \;\overrightarrow {P}_{v2j} } \\ {\left( {T_{1ix} \; \wedge \,T_{2j} } \right)\; \supseteq \;T_{v2j} } \\ {\left( {L_{1ix} \; \wedge \;L_{2j} } \right)\; \supseteq \;L_{v2j} } \\ {\left( {x_{1ix} \wedge x_{2j} } \right)\; \supseteq \;y_{v2j} } \\ {\left( {G_{U1ix} \; \wedge \;G_{U2j} } \right)\; \supseteq \,G_{v2j} } \\ { \ldots \ldots } \\ {\left( {U_{1tx} \; \wedge \;U_{2m1} } \right)\; \supseteq \,V_{2t} } \\ {\left( {S_{1tx} \; \wedge \;S_{2m1} } \right)\; \supseteq \;S_{v2t} } \\ {\left( {\overrightarrow {P}_{1tx} \; \wedge \;\overrightarrow {P}_{2m1} } \right)\; \supseteq \;\overrightarrow {P}_{v2t} } \\ {\left( {T_{1tx} \; \wedge \;T_{2m1} } \right)\; \supseteq \,T_{v2t} } \\ {\left( {L_{1tx} \; \wedge \;L_{2m1} } \right)\; \supseteq \;L_{v2t} } \\ {\left( {x_{1tx} \; \wedge \;x_{2m1} } \right)\; \supseteq \;y_{v2t} } \\ {\left( {G_{U1tx} \; \wedge \;G_{U2m1} } \right)\; \supseteq \;G_{v2t} } \\ \end{array} $$

Theorem 2

The sufficient and necessary condition for the solvability of X i A′ ⊇ B i is,

$$ \begin{array}{*{20}c} {U_{20} \; \supseteq \;V_{20} } \\ {S_{20} \; \supseteq \;S_{v20} } \\ {\overrightarrow {P}_{20} \; \supseteq \;\overrightarrow {P}_{v20} } \\ {T_{20} \; \supseteq \;T_{v20} } \\ {L_{20} \; \supseteq \;L_{v20} } \\ {x_{20} \; \supseteq \;y_{v20} } \\ {G_{U20} \; \supseteq \;G_{v20} } \\ { \ldots \ldots } \\ {U_{2j} \; \supseteq \;V_{2j} } \\ {S_{2j} \; \supseteq \;S_{v2j} } \\ {\overrightarrow {P}_{2j} \; \supseteq \;\overrightarrow {P}_{v2j} } \\ {T_{2j} \; \supseteq \;T_{v2j} } \\ {L_{2j} \; \supseteq \;L_{v2j} } \\ {x_{2j} \; \supseteq \;y_{v2j} } \\ {G_{U2j} \; \supseteq \;G_{v2j} } \\ { \ldots \ldots } \\ {U_{2m1} \; \supseteq \;V_{2t} } \\ {S_{2m1} \; \supseteq \;S_{v2t} } \\ {\overrightarrow {P}_{2m1} \; \supseteq \;\overrightarrow {P}_{v2t} } \\ {T_{2m1} \; \supseteq \;T_{v2t} } \\ {L_{2m1} \; \supseteq \;L_{v2t} } \\ {x_{2m1} \; \supseteq \;y_{v2t} } \\ {G_{U2m1} \; \supseteq \;G_{v2t} } \\ \end{array} $$

Proof 1

If one of the conditions above is not satisfied, for example suppose S 2j (t) ⊇ S v2j(t) is not satisfied, so in the (S 1ix(t) ∧ S 2j (t)) = S v2j(t), no mater what value S 1ix(t) is, we can not get (S 1ix(t) ∧ S 2j (t)) = S v2j(t)。

Proof 2

Since we could only take union operation between the corresponding element of A and Xi in the XiA′ ⊇ Bi, that is

$$ \begin{aligned} U_{1ix} & = U_{20} \; \cup \;U_{21} \; \cup \, \cdots U_{2j} \; \cup \; \cdots \; \cup \;U_{2t} \\ S_{1ix} & = S_{20} \, \cup \;S_{21} \; \cup \; \cdots S_{2j} \; \cup \; \cdots \; \cup \;S_{2t} \\ \overrightarrow {P}_{1ix} & = \overrightarrow {P}_{20} \; \cup \;\overrightarrow {P}_{21} \; \cup \; \cdots \overrightarrow {P}_{2j} \; \cup \; \cdots \; \cup \;\overrightarrow {P}_{2t} \\ T_{1ix} & = T_{20} \; \cup \;T_{21} \; \cup \; \cdots T_{2j} \; \cup \; \cdots \; \cup \;T_{2t} \\ L_{1ix} & = L_{20} \; \cup \;L_{21} \; \cup \, \cdots L_{2j} \; \cup \; \cdots \; \cup \;L_{2t} \\ x_{1ix} & = x_{20} \; \cup \;x_{21} \; \cup \; \cdots x_{2j} \; \cup \; \cdots \; \cup \;x_{2t} \\ G_{U1ix} & = G_{U20} \; \cup \;G_{U21} \; \cup \, \cdots G_{U2j} \; \cup \; \cdots \; \cup \;G_{U2t} \\ \end{aligned} $$

Then we discuss all the solutions of XiA′ ⊇ Bi and XA′ ⊇ B.

When X(x1, x2, xn) are obtained, we take intersection operation between X 与 Kg, rw, xq, so we can get X′(x′1, x′2, …, x′n) ∈ X′。

5 Examples of Error Matrix Equation

Suppose \( A^{\prime} = \left( {\begin{array}{*{20}c} {a_{11} } & {a_{12} } \\ {a_{21} } & {a_{22} } \\ \end{array} } \right) \), and

$$ \begin{aligned} a_{11} & = \left( {\begin{array}{*{20}c} {U_{201} } & {S_{201} } & {\overrightarrow {P}_{201} } & {T_{201} } & {L_{201} } & {y_{201} } & {G_{U201} } \\ \end{array} } \right) \\ a_{12} & = \left( {\begin{array}{*{20}c} {U_{202} } & {S_{202} } & {\overrightarrow {P}_{202} } & {T_{202} } & {L_{202} } & {y_{202} } & {G_{U202} } \\ \end{array} } \right) \\ a_{21} & = \left( {\begin{array}{*{20}c} {U_{211} } & {S_{211} } & {\overrightarrow {P}_{211} } & {T_{211} } & {L_{211} } & {y_{211} } & {G_{U211} } \\ \end{array} } \right) \\ a_{22} & = \left( {\begin{array}{*{20}c} {U_{212} } & {S_{212} } & {\overrightarrow {P}_{212} } & {T_{212} } & {L_{212} } & {y_{212} } & {G_{U212} } \\ \end{array} } \right) \\ \end{aligned} $$

When n = 11, we get the following equations:

$$ \begin{aligned} U_{201} & = \left\{ {u_{201} ,u_{202} , \ldots ,u_{20n} } \right\} \\ S_{201} & = \left\{ {s_{201} ,s_{202} , \ldots ,s_{20n} } \right\} \\ \overrightarrow {P}_{201} & = \left\{ {\overrightarrow {p}_{201} ,\overrightarrow {p}_{202} , \ldots ,\overrightarrow {p}_{20n} } \right\} \\ T_{201} & = \left\{ {t_{201} ,t_{202} , \ldots ,t_{20n} } \right\} \\ L_{201} & = \left\{ {l_{201} ,l_{202} , \ldots ,l_{20n} } \right\} \\ y_{201} & = \left\{ {y_{201} ,y_{202} , \ldots ,y_{20n} } \right\} \\ G_{U201} & = \left\{ {g_{201} ,g_{202} , \ldots ,g_{20n} } \right\} \\ \end{aligned} $$

When n = 9, we get the following equations:

$$ \begin{aligned} U_{202} & = \left\{ {u_{201} ,u_{202} , \ldots ,u_{20n} } \right\} \\ S_{202} & = \left\{ {s_{201} ,s_{202} , \ldots ,s_{20n} } \right\} \\ \overrightarrow {P}_{202} & = \left\{ {\overrightarrow {p}_{201} ,\overrightarrow {p}_{202} , \ldots ,\overrightarrow {p}_{20n} } \right\} \\ T_{202} & = \left\{ {t_{201} ,t_{202} , \ldots ,t_{20n} } \right\} \\ L_{202} & = \left\{ {l_{201} ,l_{202} , \ldots ,l_{20n} } \right\} \\ y_{202} & = \left\{ {y_{201} ,y_{202} , \ldots ,y_{20n} } \right\} \\ G_{U202} & = \left\{ {g_{201} ,g_{202} , \ldots ,g_{20n} } \right\} \\ \end{aligned} $$

When n = 10, we get the following equations:

$$ \begin{aligned} U_{211} & = \left\{ {u_{211} ,u_{212} , \ldots ,u_{21n} } \right\} \\ S_{211} & = \left\{ {s_{211} ,s_{212} , \ldots ,s_{21n} } \right\} \\ \overrightarrow {P}_{211} & = \left\{ {\overrightarrow {p}_{211} ,\overrightarrow {p}_{212} , \ldots ,\overrightarrow {p}_{21n} } \right\} \\ T_{211} & = \left\{ {t_{211} ,t_{212} , \ldots ,t_{21n} } \right\} \\ L_{211} & = \left\{ {l_{211} ,l_{212} , \ldots ,l_{21n} } \right\} \\ y_{211} & = \left\{ {y_{211} ,y_{212} , \ldots ,y_{21n} } \right\} \\ G_{U211} & = \left\{ {g_{211} ,g_{212} , \ldots ,g_{21n} } \right\} \\ \end{aligned} $$

When n = 15, we get the following equations:

$$ \begin{aligned} U_{212} & = \left\{ {u_{211} ,u_{212} , \ldots ,u_{21n} } \right\} \\ S_{212} & = \left\{ {s_{211} ,s_{212} , \ldots ,s_{21n} } \right\} \\ \overrightarrow {P}_{212} & = \left\{ {\overrightarrow {p}_{211} ,\overrightarrow {p}_{212} , \ldots ,\overrightarrow {p}_{21n} } \right\} \\ T_{212} & = \left\{ {t_{211} ,t_{212} , \ldots ,t_{21n} } \right\} \\ L_{212} & = \left\{ {l_{211} ,l_{212} , \ldots ,l_{21n} } \right\} \\ y_{212} & = \left\{ {y_{211} ,y_{212} , \ldots ,y_{21n} } \right\} \\ G_{U212} & = \left\{ {g_{211} ,g_{212} , \ldots ,g_{21n} } \right\} \\ \end{aligned} $$

And suppose \( X = \left( {\begin{array}{*{20}c} {x_{1} } & {x_{2} } \\ \end{array} } \right) \), where

$$ \begin{aligned} x_{1} & = \left( {\begin{array}{*{20}c} {U_{10x} } & {S_{10x} } & {\overrightarrow {P}_{10x} } & {T_{10x} } & {L_{10x} } & {x_{10x} } & {G_{U10x} } \\ \end{array} } \right) \\ x_{2} & = \left( {\begin{array}{*{20}c} {U_{11x} } & {S_{11x} } & {\overrightarrow {P}_{11x} } & {T_{11x} } & {L_{11x} } & {x_{11x} } & {G_{U11x} } \\ \end{array} } \right) \\ \end{aligned} $$

And suppose \( B^{\prime} = \left( {\begin{array}{*{20}c} {b_{11} } & {b_{12} } \\ {b_{21} } & {b_{22} } \\ \end{array} } \right) \), where

$$ \begin{aligned} b_{11} & = \left( {\begin{array}{*{20}c} {V_{201} } & {S_{v201} } & {\overrightarrow {P}_{v201} } & {T_{v201} } & {L_{v201} } & {y_{v201} } & {G_{V201} } \\ \end{array} } \right) \\ b_{12} & = \left( {\begin{array}{*{20}c} {V_{202} } & {S_{v202} } & {\overrightarrow {P}_{v202} } & {T_{v202} } & {L_{v202} } & {y_{v202} } & {G_{V202} } \\ \end{array} } \right) \\ b_{21} & = \left( {\begin{array}{*{20}c} {V_{211} } & {S_{v211} } & {\overrightarrow {P}_{v211} } & {T_{v211} } & {L_{v211} } & {y_{v211} } & {G_{V211} } \\ \end{array} } \right) \\ b_{22} & = \left( {\begin{array}{*{20}c} {V_{212} } & {S_{v212} } & {\overrightarrow {P}_{v212} } & {T_{v212} } & {L_{v212} } & {y_{v212} } & {G_{V212} } \\ \end{array} } \right) \\ \end{aligned} $$

And when k = 7, we get the following equations:

$$ \begin{aligned} V_{201} & = \left\{ {u_{201} ,u_{202} , \ldots ,u_{20k} } \right\} \\ S_{v201} & = \left\{ {s_{201} ,s_{202} , \ldots ,s_{20k} } \right\} \\ \overrightarrow {P}_{v201} & = \left\{ {\overrightarrow {p}_{201} ,\overrightarrow {p}_{202} , \ldots ,\overrightarrow {p}_{20k} } \right\} \\ T_{v201} & = \left\{ {t_{201} ,t_{202} , \ldots ,t_{20k} } \right\} \\ L_{v201} & = \left\{ {l_{201} ,l_{202} , \ldots ,l_{20k} } \right\} \\ y_{v201} & = \left\{ {y_{201} ,y_{202} , \ldots ,y_{20k} } \right\} \\ G_{v201} & = \left\{ {g_{201} ,g_{202} , \ldots ,g_{20k} } \right\} \\ \end{aligned} $$

And when k = 8, we get the following equations:

$$ \begin{aligned} V_{202} & = \left\{ {u_{201} ,u_{202} , \ldots ,u_{20k} } \right\} \\ S_{v202} & = \left\{ {s_{201} ,s_{202} , \ldots ,s_{20k} } \right\} \\ \overrightarrow {P}_{v202} & = \left\{ {\overrightarrow {p}_{201} ,\overrightarrow {p}_{202} , \ldots ,\overrightarrow {p}_{20k} } \right\} \\ T_{v202} & = \left\{ {t_{201} ,t_{202} , \ldots ,t_{20k} } \right\} \\ L_{v202} & = \left\{ {l_{201} ,l_{202} , \ldots ,l_{20k} } \right\} \\ y_{v202} & = \left\{ {y_{201} ,y_{202} , \ldots ,y_{20k} } \right\} \\ G_{v202} & = \left\{ {g_{201} ,g_{202} , \ldots ,g_{20k} } \right\} \\ \end{aligned} $$

And when k = 2, we get the following equations:

$$ \begin{aligned} V_{211} & = \left\{ {u_{211} ,u_{212} , \ldots ,u_{21k} } \right\} \\ S_{v211} & = \left\{ {s_{211} ,s_{212} , \ldots ,s_{21k} } \right\} \\ \overrightarrow {P}_{v211} & = \left\{ {\overrightarrow {p}_{211} ,\overrightarrow {p}_{212} , \ldots ,\overrightarrow {p}_{21k} } \right\} \\ T_{v211} & = \left\{ {t_{211} ,t_{212} , \ldots ,t_{21k} } \right\} \\ L_{v211} & = \left\{ {l_{211} ,l_{212} , \ldots ,l_{21k} } \right\} \\ y_{v211} & = \left\{ {y_{211} ,y_{212} , \ldots ,y_{21k} } \right\} \\ G_{v211} & = \left\{ {g_{211} ,g_{212} , \ldots ,g_{21k} } \right\} \\ \end{aligned} $$

And when k = 3, we get the following equations:

$$ \begin{aligned} V_{212} & = \left\{ {u_{211} ,u_{212} , \ldots ,u_{21k} } \right\} \\ S_{v212} & = \left\{ {s_{211} ,s_{212} , \ldots ,s_{21k} } \right\} \\ \overrightarrow {P}_{v212} & = \left\{ {\overrightarrow {p}_{211} ,\overrightarrow {p}_{212} , \ldots ,\overrightarrow {p}_{21k} } \right\} \\ T_{v212} & = \left\{ {t_{211} ,t_{212} , \ldots ,t_{21k} } \right\} \\ L_{v212} & = \left\{ {l_{211} ,l_{212} , \ldots ,l_{21k} } \right\} \\ y_{v212} & = \left\{ {y_{211} ,y_{212} , \ldots ,y_{21k} } \right\} \\ G_{v212} & = \left\{ {g_{211} ,g_{212} , \ldots ,g_{21k} } \right\} \\ \end{aligned} $$

By the Theorem 2, when n = 11, we get the solution of XA′ ⊇ B, which is

$$ \begin{aligned} U_{10x} & = \left\{ {u_{201} ,u_{202} , \ldots ,u_{20n} } \right\} \\ S_{10x} & = \left\{ {s_{201} ,s_{202} , \ldots ,s_{20n} } \right\} \\ \overrightarrow {P}_{10x} & = \left\{ {\overrightarrow {p}_{201} ,\overrightarrow {p}_{202} , \ldots ,\overrightarrow {p}_{20n} } \right\} \\ L_{10x} & = \left\{ {l_{201} ,l_{202} , \ldots ,l_{20n} } \right\} \\ x_{10x} & = \left\{ {y_{201} ,y_{202} , \ldots ,y_{20n} } \right\} \\ G_{U10x} & = \left\{ {g_{201} ,g_{202} , \ldots ,g_{20n} } \right\} \\ \end{aligned} $$

Similarly when n = 8 we get the following equations:

$$ \begin{aligned} U_{11x} & = \left\{ {u_{211} ,u_{212} , \ldots ,u_{20n} } \right\} \\ S_{11x} & = \left\{ {s_{201} ,s_{202} , \ldots ,s_{20n} } \right\} \\ \overrightarrow {P}_{11x} & = \left\{ {\overrightarrow {p}_{201} ,\overrightarrow {p}_{202} , \ldots ,\overrightarrow {p}_{20n} } \right\} \\ T_{11x} & = \left\{ {t_{201} ,t_{202} , \ldots ,t_{20n} } \right\} \\ x_{11x} & = \left\{ {y_{211} ,y_{212} , \ldots ,y_{21n} } \right\} \\ G_{U11x} & = \left\{ {g_{211} ,g_{212} , \ldots ,g_{21n} } \right\} \\ \end{aligned} $$

6 Conclusion

We get the necessary and sufficient condition for the solvability of the error matrix equation XA′ = B, and they are also can be proved by the case studies.